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Jul 20, 2013
07/13

by
David Brander

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We describe how the loop group maps corresponding to special submanifolds associated to integrable systems may be thought of as certain Grassmann submanifolds of infinite dimensional homogeneous spaces. In general, the associated families of special submanifolds are certain Grassmann submanifolds. An example is given from recent work of the author.

Source: http://arxiv.org/abs/0804.1830v1

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Sep 17, 2013
09/13

by
David Brander

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We study singularities of spacelike, constant (non-zero) mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space $L^3$. We show how to solve the singular Bj\"orling problem for such surfaces, which is stated as follows: given a real analytic null-curve $f_0(x)$, and a real analytic null vector field $v(x)$ parallel to the tangent field of $f_0$, find a conformally parameterized (generalized) CMC $H$ surface in $L^3$ which contains this curve as a singular set and such that the...

Source: http://arxiv.org/abs/0912.5081v2

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Sep 19, 2013
09/13

by
David Brander

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Let U be a real form of a complex semisimple Lie group, and tau, sigma, a pair of commuting involutions on U. This data corresponds to a reflective submanifold of a symmetric space, U/K. We define an associated integrable system, and describe how to produce solutions from curved flats. The solutions are shown to correspond to various special submanifolds, depending on which homogeneous space U/L one projects to. We apply the construction to a question which generalizes, to the context of...

Source: http://arxiv.org/abs/math/0703009v2

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Sep 23, 2013
09/13

by
David Brander

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We discuss generalizations of the well-known theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3-space. We show that this problem is expressed very naturally as the question of the existence of certain homotheties of reflective submanifolds of a symmetric space. As such, we conclude that the only other (non-compact) cases to which this theorem could generalize are the problem of isometric immersions with flat normal bundle of the hyperbolic...

Source: http://arxiv.org/abs/0710.0507v4

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Jul 20, 2013
07/13

by
David Brander

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We define a large class of integrable nonlinear PDE's, \emph{$k$-symmetric AKS systems}, whose solutions evolve on finite dimensional subalgebras of loop algebras, and linearize on an associated algebraic curve. We prove that periodicity of the associated algebraic data implies a type of quasiperiodicity for the solution, and show that the problem of isometrically immersing $n$-dimensional Euclidean space into a sphere of dimension $2n-1$ can be addressed via this scheme, producing infinitely...

Source: http://arxiv.org/abs/math/0604245v2

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Sep 22, 2013
09/13

by
David Brander

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We prove a global Birkhoff decomposition for almost split real forms of loop groups, when an underlying finite dimensional Lie group is compact. Among applications, this shows that the dressing action - by the whole subgroup of loops which extend holomorphically to the exterior disc - on the $U$-hierarchy of the ZS-AKNS systems, on curved flats and on various other integrable systems, is global for compact cases. It also implies a global infinite dimensional Weierstrass-type representation for...

Source: http://arxiv.org/abs/0805.1979v1

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Sep 19, 2013
09/13

by
David Brander

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We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms and pseudo-Riemannian space forms. As a corollary, we obtain a non-immersibility theorem for spheres into certain pseudo-Riemannian spheres and hyperbolic spaces. The second...

Source: http://arxiv.org/abs/math/0610058v3

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Sep 19, 2013
09/13

by
David Brander; Martin Svensson

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The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-space, and for timelike constant non-zero mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. We prove that there is a unique solution if the prescribed curve is...

Source: http://arxiv.org/abs/1009.5661v2

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Jul 20, 2013
07/13

by
David Brander; Josef Dorfmeister

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Let $G$ be a complex Lie group and $\Lambda G$ denote the group of maps from the unit circle ${\mathbb S}^1$ into $G$, of a suitable class. A differentiable map $F$ from a manifold $M$ into $\Lambda G$, is said to be of \emph{connection order $(_a^b)$} if the Fourier expansion in the loop parameter $\lambda$ of the ${\mathbb S}^1$-family of Maurer-Cartan forms for $F$, namely $F_\lambda^{-1} \dd F_\lambda$, is of the form $\sum_{i=a}^b \alpha_i \lambda^i$. Most integrable systems in geometry...

Source: http://arxiv.org/abs/math/0604247v5

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Sep 18, 2013
09/13

by
David Brander; Wayne Rossman

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We give a loop group formulation for the problem of isometric immersions with flat normal bundle of a simply connected pseudo-Riemannian manifold $M_{c,r}^m$, of dimension $m$, constant sectional curvature $c \neq 0$, and signature $r$, into the pseudo-Euclidean space $\real_s^{m+k}$, of signature $s\geq r$. In fact these immersions are obtained canonically from the loop group maps corresponding to isometric immersions of the same manifold into a pseudo-Riemannian sphere or hyperbolic space...

Source: http://arxiv.org/abs/math/0611033v2

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Sep 23, 2013
09/13

by
David Brander; Martin Svensson

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We use integrable systems techniques to study the singularities of timelike non-minimal constant mean curvature (CMC) surfaces in the Lorentz-Minkowski 3-space. The singularities arise at the boundary of the Birkhoff big cell of the loop group involved. We examine the behaviour of the surfaces at the big cell boundary, generalize the definition of CMC surfaces to include those with finite, generic singularities, and show how to construct surfaces with prescribed singularities by solving a...

Source: http://arxiv.org/abs/1110.4449v1

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Sep 22, 2013
09/13

by
David Brander; Josef F. Dorfmeister

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We define certain deformations between minimal and non-minimal constant mean curvature (CMC) surfaces in Euclidean space $E^3$ which preserve the Hopf differential. We prove that, given a CMC $H$ surface $f$, either minimal or not, and a fixed basepoint $z_0$ on this surface, there is a naturally defined family $f_h$, for all real $h$, of CMC $h$ surfaces that are tangent to $f$ at $z_0$, and which have the same Hopf differential. Given the classical Weierstrass data for a minimal surface, we...

Source: http://arxiv.org/abs/1302.2228v1

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Sep 22, 2013
09/13

by
David Brander; Josef F. Dorfmeister

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The classical Bj\"orling problem is to find the minimal surface containing a given real analytic curve with tangent planes prescribed along the curve. We consider the generalization of this problem to non-minimal constant mean curvature (CMC) surfaces, and show that it can be solved via the loop group formulation for such surfaces. The main result gives a way to compute the holomorphic potential for the solution directly from the Bj\"orling data, using only elementary differentiation,...

Source: http://arxiv.org/abs/0908.3274v2

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Sep 21, 2013
09/13

by
David Brander; Jun-ichi Inoguchi; Shimpei Kobayashi

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In this paper we study constant positive Gauss curvature $K$ surfaces in the 3-sphere $S^3$ with $0

Source: http://arxiv.org/abs/1301.5999v1

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Jul 20, 2013
07/13

by
David Brander; Wayne Rossman; Nick Schmitt

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We give an infinite dimensional generalized Weierstrass representation for spacelike constant mean curvature (CMC) surfaces in Minkowski 3-space $\real^{2,1}$. The formulation is analogous to that given by Dorfmeister, Pedit and Wu for CMC surfaces in Euclidean space, replacing the group $SU_2$ with $SU_{1,1}$. The non-compactness of the latter group, however, means that the Iwasawa decomposition of the loop group, used to construct the surfaces, is not global. We prove that it is defined on an...

Source: http://arxiv.org/abs/0804.1596v3